A Ternary (4k−1(2))/3 Collatz-Type Map: Complete 3-Adic Reduction, Algebraic Cycle Analysis, and Exhaustive Computational Verification up to 10^9

This paper studies a ternary Collatz-type dynamical system on the positive integers not divisible by 3. For each admissible positive integer x, let r(x) be its nonzero residue modulo 3, so that r(x) is either 1 or 2. The accelerated map is defined by T(x) = (4x − r(x)) / 3^v, where v is the exact exponent of 3 dividing 4x − r(x). Thus, all available factors of 3 are removed at every iteration. The paper develops the two residue branches of the map, exact inverse branches, the associated 3-adic valuation structure, and an algebraic cycle equation relating periodic orbits to the balance between powers of 4 and powers of 3. Three positive attractors were observed in the finite computation: the fixed points 1 and 2, together with the four-cycle 22 -> 29 -> 38 -> 50 -> 22. An exhaustive computational verification was performed for every admissible positive starting value x <= 10^9, comprising 666,666,667 starting values. Every tested orbit reached one of the three observed attractors. No additional terminal cycle and no arithmetic overflow were encountered. The basin counts were: Fixed point 1: 66,019,616 starting values. Fixed point 2: 432,100,455 starting values. Four-cycle 22 -> 29 -> 38 -> 50 -> 22: 168,546,596 starting values. The longest observed transient required 405 iterations of the accelerated map and occurred for the starting value 833,560,375. The same starting value accumulated 529 total removed factors of 3 before reaching its terminal cycle. The largest fully reduced state observed was 164,210,561,845,359,809, occurring along the orbit starting from 664,934,564. The paper also develops a 3-adic heuristic under which the expected number of removed factors of 3 per accelerated step is 3/2. This gives a negative average logarithmic drift. The drift calculation is heuristic and is not a proof of global convergence. Motivated by the algebraic structure, the 3-adic heuristic, and the exhaustive finite computation through 10^9, the paper proposes the positive three-attractor convergence conjecture: every positive integer not divisible by 3 eventually reaches 1, 2, or the cycle 22 -> 29 -> 38 -> 50 -> 22. The exhaustive computation through 10^9 provides finite computational evidence for this conjecture. It does not constitute a proof for all positive integers. The accompanying archive contains the computational material used for reproducibility, including source code, computational scripts, all 100 computational chunks covering the verification through 10^9, merged global results, the LaTeX source, and the article PDF.